As mentioned above, we are interested in finding the zeros of the electric field. These are defined as points \({x^*}\in \mathbb {R}^3\)  such that \({E}({x^*})= {0}\) . To find them, we start with initial guesses and then refine the guesses iteratively, using a multivariable Newton method. To obtain reasonable initial guesses, we use an algorithm known as “3D Marching Cubes,” a computer graphics algorithm for finding level sets of a scalar function [1]}. Here, we use Marching Cubes on each of the three components of \({E}({x})\)  to find their zero level sets. The algorithm partitions a large cuboid containing the knot into small cubes; then, on each cube, it uses a bilinear approximation of that component. If the bilinear approximations of all three components of \({E}\)  pass through zero in the same cube, then we take the center of that cube as an initial guess for our multivariable Newton method. Some initial guesses to Newton end up diverging or converging to a far away critical point, and we throw these away. In contrast, the successful initial guesses quickly converge to the approximate locations of the zeros of \({E}({x})\) .
